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Accepted Manuscript
A multiscale modelling approach for the regulation of the cell cycle by the circadian clock
Raouf El Cheikh, Samuel Bernard, Nader El Khatib
PII: S0022-5193(17)30229-1
DOI: 10.1016/j.jtbi.2017.05.021
Reference: YJTBI 9078
To appear in: Journal of Theoretical Biology
Received date: 9 September 2016
Revised date: 16 May 2017
Accepted date: 17 May 2017
Please cite this article as: Raouf El Cheikh, Samuel Bernard, Nader El Khatib, A multiscale modelling approach for the regulation of the cell cycle by the circadian clock, Journal of Theoretical Biology
(2017), doi:10.1016/j.jtbi.2017.05.021
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Highlights
• A multiscale model for the cell cycle-circadian clock coupling is developed
• The model is based on a transport PDE structured by molecular contents
• A particle-based method is used for resolution
• Impacts of inter and intracellular dynamics on cell proliferation are studied
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A multiscale modelling approach for the regulation of
1
the cell cycle by the circadian clock
2
Raouf El Cheikh
1, Samuel Bernard
2and Nader El Khatib
∗33
1Aix Marseille Univ, Inserm S 911 CRO2, SMARTc Pharmacokinetics Unit, 27 Bd Jean Moulin,
4
Marseille, France
5
2CNRS UMR 5208, Institut Camille Jordan, Universit´e Lyon1, 43 blvd. du 11 novembre 1918,
6
F-69622 Villeurbanne cedex, France
7
3Lebanese American University, Department of Computer Science and Mathematics, Byblos, P.O.
8
Box 36, Byblos, Lebanon
9
May 23, 2017
10
Abstract
11
We present a multiscale mathematical model for the regulation of the cell cycle by
12
the circadian clock. Biologically, the model describes the proliferation of a population
13
of heterogeneous cells connected to each other. The model consists of a high
dimen-14
sional transport equation structured by molecular contents of the cell cycle-circadian
15
clock coupled oscillator. We propose a computational method for resolution adapted
16
from the concept of particle methods. We study the impact of molecular dynamics
17
on cell proliferation and show an example where discordance of division rhythms
be-18
tween population and single cell levels is observed. This highlights the importance of
19
multiscale modeling where such results cannot be inferred from considering solely one
20
biological level.
21
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1
Introduction
22
The mammalian cell cycle and the circadian clock are two molecular processes that
23
operate in a rhythmic manner. On one hand, the cell cycle is driven by the rhythmic
24
activity of cyclin-dependent kinases, which dictate the time a cell must engage mitosis
25
and the time it must divide giving birth to two daughter cells. On the other hand, the
26
circadian clock is a network of transcriptional and translational feedback-loops that
27
generate sustained oscillations of different mRNAs and proteins concentration with a
28
period of approximately 24h. It turns out that several components of the
circadian-29
clock regulate various cyclin-dependent kinases at different stages of the cell cycle. This
30
makes the circadian clock a key player in the temporal organization of the cell cycle
31
and makes these two biological processes act as two tightly coupled oscillators.
32
Many computational models explored the interaction between the cell cycle and
33
the circadian clock. One category of models consists of systems of ordinary differential
34
equations that describe the interaction at the molecular level [14, 33]. The second
cat-35
egory consists of physiologically-structured partial differential equations that describe
36
the effect of circadian regulation on the growth rate of cells [8, 9, 2, 7]. In a recent
37
study, we have combined these two approaches by coupling an age-structured PDE to
38
a circadian clock-cell cycle molecular ODE system [12]. Although this model could
39
capture the influence of intracellular dynamics on the growth rate of cells, it lacked
40
a proper multiscale description [1]. It could not take into consideration, for example,
41
intracellular heterogeneity among cells.
42
Here, we present a multiscale formulation of our mathematical model for the
reg-43
ulation of the cell cycle by the circadian clock and a numerical method for solving it.
44
The multiscale formulation consists of a transport equation structured by the
molecu-45
lar contents of the coupled cell cycle-circadian clock oscillator. In its general form, the
46
equation reads
47
∂tρ(x, t, λ) +∇x·u(x, t, λ, ψ)ρ(x, t, λ)= L[x, λ] ρ(x, t, λ), (1.1)
with ρ representing the density of cells. In the coupled PDE/ODE model, the PDE
48
(equation (11) in [12]) was structured in age. By contrast, partial derivatives in
equa-49
tion (1.1) are structured by the molecular contents (denoted x) and the space where
50
it is solved is d-dimensional with d the number of molecular components xi; this is
51
why we call it a molecular-structured equation. Since the molecular mechanism of
52
the cell cycle-circadian clock interaction involves an abundant number of components,
53
the model is high-dimensional in nature. Classical numerical methods such as finite
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volumes/differences are inappropriate for solving the transport equation. This is the
55
main difficulty that makes molecular-structured models scarcely used in similar
appli-56
cations. We circumvent this difficulty by adapting a particle method to solve the main
57
equation.
58
Particle methods have arisen as an alternative to classical numerical methods for
59
solving high-dimensional problems. They are used in different applications, like the
60
incompressible Euler equations in fluid mechanics [17, 19, 20], the Vlasov equation
61
in plasma physics [3, 15] and in turbulence models for reactive flows [25, 24]. The
62
computational implementation of particle methods is conceptually simple. At a given
63
time, the solution is represented by a large number of particles, each having its own
64
properties, for example position and weight. These properties evolve in time according
65
to a system of stochastic or ordinary differential equations. The classical solution to the
66
PDE is obtained by reconstructing a smooth density from the particles distribution.
67
Theoretically, the numerical solution is a linear combination of Dirac masses
68 ρ(x, t) = N X j=1 αj(t)δ(x− xj)
where αj is the weight of particle j, xj its position (state) and N is the total number
69
of particles. To obtain a classical solution, one has to update the positions and weights
70
of particles and then regularize the Dirac masses. The overriding strength of particle
71
methods is that, for N fixed, the size of the system increases only linearly with the
72
dimensionality of the space. This means that if we have a structured equation with
73
dimension d, we have to solve d× N ODEs/SDEs. Since recovering a classical solution
74
requires a regularization of the Dirac distribution, the performance of particle methods
75
depends also on the quality of the regularization procedure. For generalities about
76
particle methods, one can refer to [23, 27, 10].
77
In this work, we investigated the influence of molecular dynamics on
78
cell proliferation. Therefor, the novelty of our approach emanate from
79
considering a multiscale design that incorporates three linked biological
80
levels. Hereafter some specificities of our model:
81
• Intracellular heterogeneity among cells: generally speaking, the particle method
82
consists of solving N copies of an ODE/SDE system, each representing
83
the dynamics of a given particle. In our model, the N copies are not
84
the same. Since each particle is associated with a given cell, we
consid-85
ered several sources of variability. For example, we have multiplied the
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right-hand side of the ODE system describing cell cycle dynamics by
87
a factor λ. This ensures that each cell has a distinct cell cycle period.
88
Another source of variability was introduced with cell division. Once
89
a cell divides, its daughter is attributed a new coefficient λ given by
90
λN +1= λN+ Dξλ, with D a ”diffusion” coefficient and ξλ a normally
dis-91
tributed random number. This implies that the daughter and mother
92
cells will have distinct cell cycle periods.
93
• Inter-cellular connection: we assumed that cells are connected with each
94
other by making Per/Cry mRNA transcription dependent on the
av-95
erage concentration of Per/Cry among cells. Including connectivity
96
between cells is important in certain tissues to maintain a robust
syn-97
chronized activity.
98
• Coupling back from population to molecular level: we assumed that cell
di-99
vision is dependent on the MPF-WEE1 molecular activity and that
100
the total number of cells has an impact on Per/Cry mRNA activity.
101
This implies a two way coupling between the molecular and population
102
levels. We assumed also that the MPF activation coefficient decreases
103
at an exponential rate proportional to the total number of cells. This
104
ensures a limited growth, which is physiologically more realistic.
105
The paper is divided mainly into two parts. First, we introduce the model structure,
106
give details about its construction and how the transport equation is solved by adapting
107
a particle method. Then, we illustrate biological properties that can be examined
108
using our model; like the effect of intracellular dynamics on cell proliferation, the
109
synchronization among cells, and how heterogeneity among cells affects growth rate.
110
2
Description of the model
111
2.1
General framework and main equation
112
We consider a large collection of cells with state (x, λ)∈ Ω × Λ ⊂ Rd× Rp, where x is a
113
cellular dynamical state in an open subset Ω of dimension d and λ a cellular parameter
114
state in an open subset Λ of dimension p. In our case, x = x1,· · · , xd with d = 10
115
represents the set of proteins/mRNAs concentrations and λ∈ R (p = 1) represents the
116
intrinsic cell cycle period of each cell. The distinction between the dynamical and the
117
parameter state is that the dynamical state changes during the lifespan of a cell, while
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the parameter state is fixed over the lifespan of the cell, but can vary among cells or
119
when a cell divides. The population is described by its density ρ(x, t, λ) which evolves
120
according to the following hyperbolic PDE
121 ∂ρ ∂t(x, t, λ) +∇x· u(x, t, λ, ψ)ρ(x, t, λ)= 1 2∆λ· σ2(x, t, λ)R(x, λ)ρ(x, t, λ)+ r(x, λ)ρ(x, t, λ) | {z } L[x, λ] ρ(x, t, λ) . (2.2)
We assumed that the cellular state dynamics x is governed by a deterministic system of ODEs, represented by the term u(x, t, λ, ψ), which depends on the cellular state (x,λ) and on m population-level statistics ψ : Ω× R × Λ → Rm where
ψi=hρ, Φii =
ZZ
Ω×Λ
ρ(x, t, λ)Φi(x, t, λ)dxdλ, i = 1,· · · , m (2.3)
with Φi: Ω×R×Λ → R, i = 1, · · · , m taken such that ψiis finite. The operator L[x, λ]
122
describes the population relative growth rate. The terms σ, R and r are diffusion in
123
parameter state, differentiation and growth rates of the population, respectively. For
124
the sake of simplicity, we omitted the full description of the derivation of equation (2.2).
125
The reader could refer to Supplementary materials for more details. We adopted a more
126
computational and biological description of the problem.
127
2.2
Intracellular dynamics
128
The evolution of cells in the state space depicts the cell cycle-the circadian clock
cou-129
pling mechanism. Therefore, intracellular dynamics were described according to the
130
following deterministic ODE system presented in [12]:
131
dx
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where u1= dx1 dt = ν1b(x7+ c) k1b(1 + (kx1i3)p) + x7+ c− k1dx1+ ksψ1, (2.5) u2= dx2 dt = k2bx q 1− k2dx2− k2tx2+ k3tx3, (2.6) u3= dx3 dt = k2tx2− k3tx3− k3dx3, (2.7) u4= dx4 dt = ν4bxr3 kr 4b+ xr3 − k4dx4, (2.8) u5= dx5 dt = k5bx4− k5dx5− k5tx5+ k6tx6, (2.9) u6= dx6 dt = k5tx5− k6tx6− k6dx6+ k7ax7− k6ax6, (2.10) u7= dx7 dt = k6ax6− k7ax7− k7dx7, (2.11) u8= dx8 dt = λ klmpf+k0mpfexp−ηψ2 kn 1mpf kn 1mpf+xn8+sxn10 (1− x8)− dwee1x9x8 , (2.12) u9= dx9 dt = λ k actw kactw+ dw1(cw+ Cx7) + kactw kactw+ dw1 − 1 kinactwxn8x9 kn1wee1+ xn8 − dw2x9 , (2.13) u10= dx10 dt = λ kact(x8− x10) . (2.14)Equations (2.5-2.11) describe the circadian oscillator and equations (2.12-2.14) the
132
cell cycle. Circadian dynamical variables are x1, Per2 or Cry mRNA and proteins;
133
x2, PER2/CRY complex (cytoplasm); x3, PER2/CRY complex (nucleus); x4, Bmal1
134
mRNA; x5, BMAL1 cytoplasmic protein; x6, BMAL1 nuclear protein and x7, active
135
BMAL1. Cell cycle dynamical variables are x8, active MPF; x9, active WEE1 and
136
x10, active MPF inhibitor. The coupling between these two oscillators is taken into
137
consideration in the term Cx7 of equation (2.13) through the regulation of WEE1 by
138
BMAL1/CLOCK. The coefficient C is the coupling strength. Details about this model
139
can be found in [12]. The cellular dynamics is now part of a multiscale system, some
140
changes were added to the original system to reflect that cells interact with each other.
141
These changes are detailed in the coming paragraphs.
142
2.3
Population synchronization
143
Including connectivity between cells is important in certain tissues to maintain a robust
144
synchronized activity. It is known, for example, that neuronal clocks within the the
145
suprachiasmatic nucleus SCN form a heterogeneous network that must synchronize to
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maintain time keeping activity. The coherent output of the SCN is established by
147
intracellular signaling factors, such as vasointestinal polypeptide [13]. A simple way to
148
induce synchronization in our model is to make Per/Cry mRNA transcription depends
149
on the average concentration of Per/Cry among cells. For that, we included in equation
150
(2.5) the term ksψ1 := kshρ, Φ1i with ks a coefficient that represents the connectivity
151
strength and Φ1 = x1. Although peripheral tissues, as modeled here, would run
out-152
of-phase without the SCN, there are so many factors that can synchronize clocks that
153
it is relevant to study the possibility for peripheral tissues to be connected.
154
2.4
Variability among cells
155
Variability among cells arises naturally from the difference in their molecular contents.
156
This was taken into consideration in our model through the initial states of cells,
157
which can be chosen randomly. We added another source of variability through the
158
parameter λ. We assumed that each cell has a distinct cell cycle period. This was
159
modeled by multiplying equations (2.12-2.14) by a scaling factor λ. Cell division can
160
also induce variability by assigning to each new born cell an intrinsic cell cycle period
161
that is different from its mother cell. (This is accomplished by taking a non trivial
162
distribution p(λ|z), see Supplementary materials equations (1.4), (1.5)).
163
2.5
Cell division
164
A central aspect of our model is that it takes into account cell division. Cell division
165
timing is based on the antagonistic relationship between the mitosis promoting factor
166
MPF and the protein WEE1. It is assumed that a cell enters mitosis once the activity of
167
MPF surpasses that of WEE1 and then divides once MPF activity shuts down abruptly.
168
Based on this mechanism, we considered two division types, one deterministic and one
169
stochastic. The deterministic division occurs exactly every time MPF activity rises
170
above WEE1 activity and then shuts down. This was taken into consideration with
171
a growth rate r(x, λ) := rd(x, λ) = δ(x∈Γ). For the stochastic division, we considered
172
that a cell divides with a certain probability4t × r(x, λ) + o(4t) for a small time step
173
4t. The division rate r is a function of x and λ that mimics the deterministic case.
174
For example, a switch-like function that takes small values on one side of Γ and large
175
values on the other side could be used (see Supplementary materials for more details
176
about the growth rates and Γ). Here, we used the Koshland-Goldbeter function [16]
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given by 178 K(y, z) = 2yJi z− y + zJa+ yJi+p(z− y + zJa+ yJi)2− 4yJi(z− y) . (2.15) This function generates a switching behavior [21]. If the ratio y/z (y and z are dummy179
variables here) becomes larger than one, the function switches to the upper state and
180
the transition occurs. Ja and Ji are two constants that determines the stiffness of the
181
switch, if they converge to zero, the switch converges to a step function.
182
2.6
Limited growth
183
To make sure the growth is bounded (for physiological and computational reasons),
184
we assumed that cell proliferation slows down when the total number of cells reaches
185
a threshold value. We assumed that the activation coefficient of MPF decreases at
186
an exponential rate proportional to the total number of cells. This was taken into
187
consideration in equation (2.12) by multiplying MPF activation coefficient k0mpf by
188
exp−ηψ2 where ψ
2 = hρ, Φ2i with Φ2 = 1 (ψ2 = N (t)). When cell number is large
189
enough, MPF activity cannot increase above that of WEE1, and cell division is blocked.
190
2.7
Method of resolution
191
The main difficulty for solving equation (2.2) comes from the high dimension of the
192
dynamical state space (10 in our case). For that, we adapted a particle method that
193
is more appropriate in high dimensions than classical methods such as finite
differ-194
ences/volumes.
195
The intuition behind the particle method is to start with a distribution of particles
196
that approximates the initial condition and then follow the evolution of these particles
197
in time in the dynamical and parameter states space. Mathematically, the particle
198
solution is a discrete measure function, which means it is irregular. To obtain a solution
199
in the usual classical sense at a given time T , one has to recover the classical solution
200
with regularization techniques. The method is mainly divided into two steps
201
2.7.1
Step 1: approximation of the initial condition
202
Given an initial condition ρ0(x, λ)∈ C0(Rd× R), we take an initial set of particles xj
203
with weights αj such that ρ0h= N
X
j
αjδ[(x, λ)− (xj, λj)] approximates ρ0. This should
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be understood as an approximation in the sense of measures, which means that one
205
looks for a test function φ∈ C0
0(Rd× R) such that 206 hρ0, φi = Z Rd ρ0φdxdλ andhρ0h, φi =X j αjφ(xj, λj).
This yields a typical numerical quadrature problem where the integral Z Rd ρ0φ is ap-207 proximated by N X j
αjφ(xj, λj). For biological purposes, in our study we do consider
208
directly an initial condition of the form (αj = 1)
209 ρ0h= N X j δ[(x, λ)− (xj, λj)].
2.7.2
Step 2: particle evolution in time
210
In the dynamical state space the particles positions Xj(t) (not to confuse between Xj
211
and xj, our notations implies that Xj(0) = xj) can be computed at a given time t by
212
solving the following ordinary differential equation system
213
d dtX
j(t) = u(Xj(t), t). (2.16)
To take into account the source term in equation (2.2), we examined the intracellular state of each cell (position of X in the state space) after each time step4t and looked if the biological conditions are met for a cell to divide. This means that if for a given particle, MPF activity rises above that of WEE1 and then shuts down under a threshold level, we add a particle XN +1 with intracellular state similar to the mother cell or randomly chosen. If division occurs, a new parameter λN +1 = λN + Dξλ is
assigned to the particle of number N + 1, with D a diffusion coefficient and ξλ a
normally distributed random number. Hence the measure solution at a given time T , is given by ρh(x, T ) = ˜ N X j δ[(x, λ)− (Xj(T ), λj(T ))]. (2.17)
with ˜N the new number particles.
214
Commonly, with particle method one needs to obtain a solution in the
215
classical sense, which means that the particle solution (2.17) should be
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regularized. This is usually accomplished by taking the convolution product
217
of ρh with a smoothing kernel ζ. In our case, we did not need to obtain
218
a smooth solution, since there is no explicit dependence of the quantities
219
of interest on the density of cells ρ. We focused on studying statistic-like
220
properties of the distribution of cells. Hence, we did not perform any
221
regularization procedure on the particle solution. For more information about
222
particle methods the reader could refer to [24, 27].
223
3
Results
224
3.1
Regulation of the cell cycle by the circadian clock
225
We solved equation (2.2) with L = 0, i.e. without division. We assumed that each cell
226
has an intrinsic cell cycle period included in the interval [20 h, 28 h] . This was done
227
by assigning to each particle a different coefficient λ. We let all cells have the same
228
initial molecular state and considered two cases; one without coupling to the circadian
229
clock (C = 0), and one with coupling to the circadian clock (C = 1.2). We plotted a
230
projection of the solution in the subspaces (x7, x9) and (x7, x8), which are the subspaces
231
of molecular concentrations (BMAL1/CLOCK, WEE1) and (BMAL1/CLOCK, MPF).
232
We obtained a random distribution of particles for the non-coupled case and a limit
233
cycle distribution for the coupled case (Figure 1). This means that the circadian clock
234
is forcing all cells to oscillate with the same period. This is in agreement with results
235
obtained earlier [12]; it was shown that a population composed of cells with cell cycle
236
periods between 20 h and 28 h are brought to oscillate with a unique cell cycle period
237
of 24 h for large values of C.
238
3.2
Cell division
239
We took four cells initially, two of them with λ = 230, and two with λ = 180 (intrinsic
240
cell cycle lengths' 10 h and 12.7 h respectively). These values were taken randomly.
241
We solved equation (2.2) with a source term and considered three cases, one with r = rd
242
(deterministic growth) and the two other ones with r = rs(stochastic growth). For the
243
stochastic growth, we used a stiff and a non-stiff Koshland-Goldbeter function. First,
244
we fixed the value of η to 0, which means that MPF activity does not depend on the
245
total cells number. We assumed that newborn cells retain the cell cycle length of their
246
mother cells at the division time, hence p(λ|z) = δ(λ − z) which leads the growth term
247
to ne L = r(x, λ)ρ. This ensures that MPF cycle does not change along birth. Our
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A
0.8 0.9 1 1.1 1.2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 BMAL1/CLOCK WEE1B
0.8 0.9 1 1.1 1.2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 BMAL1/CLOCK WEE1C
0.8 0.9 1 1.1 1.2 0.0 0.1 0.2 0.3 BMAL1/CLOCK MPFD
0.8 0.9 1 1.1 1.2 0.0 0.1 0.2 0.3 BMAL1/CLOCK MPFFigure 1: Solution of equation (2.2) without cell division. (A,B) Projection on the subspace
(x
7, x
9); (C,D) projection on the subspace (x
7, x
8). (A,C) Without coupling to the circadian
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simulations showed that the total number of cells after four days was equal to 1152 for
249
the first case, 900 for the second one with a stiff Koshland-Goldbeter function and 480
250
with a non stiff one. This is natural and is justified by the fact that the division rate rd
251
is a Dirac-type division which means that division depends deterministically on MPF
252
activity. Looking at MPF activity, we see that it peaks 9 times for λ = 230, and 6 times
253
for λ = 180 every 4 days (Figure 2A,B). If division occurs exactly once MPF activity
254
accomplishes a normal cycle, the total number of cells should be 2× 29+ 2× 26= 1152,
255
which is the result obtained with r = rd (Figure 2C). The division rate rs introduces
256
stochasticity in the decision for division. In this case, division does not depend only on
257
MPF-WEE1 activity, but also on the probability rs× 4t at each time step 4t Figure
258
(2C).
259
Second, we assumed that proliferation depends on the total number of cells, and
260
set η = 2× 10−3 (increasing the value of η will make the factor exp−ηψ2 less than one 261
and hence decreases the value of MPF activation coefficient k0mpf). We took initially
262
100 cells with different intrinsic cell cycle periods and followed the total number of cells
263
over 15 days. We remarked that division stopped after 8 days, and the growth curve
264
reached a steady state (Figure 2D). Increasing the value of η decreases the activity of
265
MPF with increasing number of cells. This means that at a certain time, MPF activity
266
will decrease below a threshold that does not allow the cell to start mitosis.
267
3.3
Synchronization
268
To study the synchronization of cells, we compared the molecular concentrations of a
269
cell chosen randomly with the average molecular concentration of all cells. We let the
270
rate of production of Per2/Cry mRNA of each cell to be dependent on the average
271
value for Per2/Cry mRNA. This was done by taking ks = 0.05 in equation (2.5). We
272
took a population of 100 cells with autonomous cell cycle periods randomly distributed
273
between 20 and 28 hours. Initial molecular concentrations were chosen randomly
be-274
tween 0 and 1 and each cell has a distinct initial molecular state. We did not consider
275
division hence L = 0, but we considered coupling between the cell cycle and the
cir-276
cadian clock (C = 1.2). We followed the evolution of three components, Per2/Cry
277
mRNA, BMAL1/CLOCK and WEE1, over 20 days. Our simulations showed that for
278
ks = 0, the average molecular concentration had small oscillations which are
asyn-279
chronous with those of the random cell concentration (Figure 3 A,B,C). Whereas for
280
ks= 0.05, we obtained that the average molecular concentration tends to coincide with
281
that of the random cell (Figure 3 D,E,F). This indicates that all cells are oscillating in
282
a similar manner (same period and phase), and indicates synchronization of all
ACCEPTED MANUSCRIPT
A
0 0.5 1 1.5 2 2.5 3 3.5 4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Days Mol. conc. [nM] MPF WEE1 BMAL1/CLOCKFirst division Second division No division
B
0 0.5 1 1.5 2 2.5 3 3.5 4 0.0 0.1 0.1 0.2 0.2 0.3 0.3 Days MPF [nM] λ = 230 λ = 180C
0 0.5 1 1.5 2 2.5 3 3.5 4 0.0 200.0 400.0 600.0 800.0 1000.0 1200.0 Days Number of cellsDirac division type Stiff Koshland−Goldbeter Non stiff koshland−Goldbter
D
0 5 10 15 0.0 200.0 400.0 600.0 800.0 1000.0 1200.0 1400.0 Days Number of cells Dirac division Koshland−Goldbeter divisionACCEPTED MANUSCRIPT
lation cells. In the case ks = 0, even though cells are coupled in the same manner to
284
their circadian clock, cells keep oscillating in an asynchronous manner. The circadian
285
clock regulates all cells to oscillate in a similar period which is 24 h in this case but
286
due to the randomness in the initial molecular concentrations, each cell oscillates with
287
a different phase.
288
3.4
Heterogeneity of cells and growth rate
289
We studied the growth rate of a cell population where each cell had a different cell
290
cycle period. We took 100 cells initially and conferred to each of them a coefficient λ
291
chosen randomly between 128 and 193. These coefficients scale the intracellular cell
292
cycle system so that periods range randomly from 12 h to 18 h. The circadian control
293
strength value C was fixed to 1.6. We did not consider connection between cells (ks= 0)
294
and we did not consider dependence of the intracellular dynamics on any population
295
level statistics. Simulations were done over 20 days and with a net death rate to mainly
296
reduce the computation time. Simulations showed a growth rate with two daily peaks,
297
suggesting that there were two division rounds every day (Figure 4A). To explain this
298
bimodal behavior for the growth rate, we looked at the average activity of WEE1 and
299
MPF. Simulations showed that MPF activity overcomes that of WEE1 once a day
300
(Figure 4B). Based on average MPF activity alone, division should occur only once
301
a day, and cannot explain the two daily peaks. We then followed a subpopulation of
302
cells and examined at which time each cell divided. We identified three regimes for
303
division: a single division per day, two divisions per day and three divisions every
304
two days (Figure 4C,D). These regimes can be explained by the entrainment results
305
obtained previously (for autonomous cell cycle periods between 12 and 18 h) showing
306
large phase-locking regions entrainment with ratios 1:1, 2:1 and 3:2 [12]. The double
307
peaks for the growth rate can be then explained by the fact that all cells are dividing
308
at least once a day and some of them will undergo a second round of division (Figure
309
4D).
310
4
Discussion and conclusion
311
We presented in this work a multiscale mathematical model for the regulation of the
312
cell cycle by the circadian clock and its influence on cell growth. The model consisted
313
of a master transport partial differential equation of high dimension structured by the
314
molecular contents of the coupled cell cycle-circadian clock oscillator. We proposed a
ACCEPTED MANUSCRIPT
A
0 2 4 6 8 10 12 14 16 18 20 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Days Per2/Cry mRNA Average value Random cellD
0 2 4 6 8 10 12 14 16 18 20 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Days Per2/Cry mRNA Average value Random cellB
0 2 4 6 8 10 12 14 16 18 20 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Days BMAL1/CLOCK Average value Random cellE
0 2 4 6 8 10 12 14 16 18 20 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Days BMAL1/CLOCK Average value Random cellC
0 2 4 6 8 10 12 14 16 18 20 0.0 0.5 1.0 1.5 2.0 2.5 Days WEE1 Average value Random cellF
0 2 4 6 8 10 12 14 16 18 20 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Days WEE1 Average value Random cellACCEPTED MANUSCRIPT
A
0 1 2 3 4 5 6 7 8 9 10 0.0 0.1 0.2 0.3 0.4 Days Growth rateB
0 1 2 3 4 5 6 7 8 9 10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Days Mol. conc. [nM] MPF avg WEE1 avgC
0 1 2 3 4 5 6 7 8 9 10 −1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.011.0 Divisions count per day
Days Cells index
D
10 11 12 13 14 15 16 17 18 19 20 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0Divisions count per day
Days
Cells index
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computational approach for resolution adapted from the concept of particle methods.
316
Several computational and theoretical studies dealt with modeling cell population
317
connected to molecular systems influencing cell growth. Bekkal Brikci and colleagues,
318
developed a nonlinear model for the dynamics of a cell population divided into a
pro-319
liferative and quiescent compartments. This model is structured by the time spent by
320
a cell in the proliferative phase and by the Cyclin D CDK4/6 amount [5, 6]. Ribba
321
and colleagues developed a multiscale model of cancer growth based on the genetic
322
and molecular features of the evolution of colorectal cancer [28]. In more recent works,
323
Prokopiou et al. presented a multiscale computational model to study the maturation
324
of CD8 T-cells in a lymph node controlled by their molecular profile [26]. Schluter
325
and colleagues developed a multiscale multicompartment model that accounts for
bio-326
physical interactions on both the cell-cell and cell colonies levels [29]. Bratsun and
327
colleagues proposed a multiscale model of cancer tumor development in epithelial
tis-328
sue. Their model includes mechanical interactions, chemical exchange as well as cell
329
division [4]. Shokhirev and colleagues developed a multiscale model to study the role of
330
NF-κB in cell division. Their approach allows for the prediction of dynamic organ-level
331
physiology in terms of intracellular molecular network.
332
All of these works emphasize on the importance of taking into account different
333
levels of a biological system by showing results that cannot be inferred by considering
334
solely one level of functioning. We exposed a simulation showing discordance between
335
rhythms observed on the population and signal cell levels. This is important since it
336
underlies the fact that biological function markers which most of the times rely on
337
average-like information can mislead the interpretation in some cases. Our modeling
338
approach differs from the above cited works which can be separated into two
cate-339
gories. One consisted of low dimension structured partial differential equations solved
340
by classical numerical techniques [28, 5]; and the other one consisted of agent-based
341
modeling approaches [4, 29, 30]. Our approach consists of considering a master
par-342
tial differential equation in high dimension structured by molecular contents for which
343
we propose a computational solution based on a particle method. This allowed us to
344
structure the partial differential equation by a large number of molecular contents,
345
providing a realistic representation of the coupled cell-cycle circadian clock oscillator.
346
By doing so, we were able to describe both population and single cell functioning in
347
one equation. Globally, the model takes into consideration the molecular dynamics
348
inside a cell, connection between heterogeneous cells and the proliferation of cells.
349
Our results can shed light on the superiority of frequent administration of
chemother-350
apies over the standard maximum tolerated dose (MTD). Scheduling the MTD into
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multiple administrations per cycle proved to be more effective and less toxic [18, 22, 31].
352
We showed in section 3.4 that an apparent population growth rhythm may hide
sev-353
eral subpopulation growth rhythms. Biological markers, as the Ki-67 index, are mostly
354
indicative of the growth status at the population level. Therefore, one is tempted to
355
consider that the administration of an anticancer drug during the time of the largest
356
growth peak is the most effective. However, our simulations showed that this largest
357
growth peak is obtained from the crossing of different subpopulation rhythms.
There-358
fore, administrating a drug at the same time every treatment cycle may hit only one
359
subpopulation of cells, leaving another subpopulation without treatment for a long
360
time. This possibly leads to the emergence of resistant subpopulations and make the
361
treatment ineffective.
362
The aim of this work was to provide a computational multiscale framework for the
363
regulation of the cell cycle by the circadian clock. The use of multiscale computational
364
tools is of growing interest in the area of drug development, especially the area of
sys-365
tems pharmacology. A major challenge for systems pharmacology is the development
366
of mechanistic tools to understand how regulatory networks control variability in drug
367
response at the organismal level [34]. Our model was constructed in the hope to
over-368
come such challenges. It had the specificity of describing the connection between the
369
cell cycle and the circadian clock through the regulation of the protein kinase WEE1 by
370
the complex BMAL1-CLOCK. Irregular activity of the WEE1 kinase has been linked to
371
ovarian and NSCLC cancers. Inhibiting this activity demonstrated interference with
372
DNA damage response within tumor cells, leading to their death. Drugs targeting
373
WEE1 are currently under investigation [11]. In such specific applications, our model
374
can be of great utility. According to Zhao and colleagues, experimental settings in
375
the process of drug development are sometimes not sufficient due to the multitude of
376
interactions between mammalian network proteins. Actions on the desired targets may
377
lead to adverse events occurring at another level due to the wide ramification of
molec-378
ular interactions. The abundance of such interactions makes it hard to design drugs
379
affecting only the desired targets with controlled therapeutic effects [34]. Therefore it
380
is important to propose modeling frameworks that allow in silico experimentation in
381
the presence of uncertain or incomplete data. In this manuscript, we have provided
382
such an example, and showed why incomplete population level data may lead to a
383
suboptimal treatment.
384
Multiscale modeling in systems medicine is growing in importance. Wolkenhauer
385
and colleagues recently proposed different recommendations to improve the integration
386
of multiscale models into clinical research [32]. One of their medium-term
ACCEPTED MANUSCRIPT
dations is the rising need of developing computational tools and algorithms for efficient
388
multiscale simulations. We hope the modeling approach we propose in this study
rep-389
resents a step forward towards this direction.
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